Question

1. (overline{ac}) is the perpendicular bisector of (overline{gh}). determine the length of the following sides. (lesson 4.4) (1 point)

a. (overline{gh})
b. (overline{ch})
c. with the information shown above, we can also determine that (overline{bc}congoverline{bc}). therefore, which of the following theorems affirms that (\triangle gcbcong\triangle hcb)? choose two.
a side - side congruence.
b angle - side - angle congruence.
c side - angle congruence.
d side - angle - side congruence.

Explanation:

Step1: Use the property of perpendicular bisector

Since \(AC\) is the perpendicular bisector of \(GH\), then \(GB = HB\) and \(\angle GBC=\angle HBC = 90^{\circ}\), and \(BC = BC\) (common side).

Step2: Apply the Hypotenuse - Leg (HL) congruence theorem for right - triangles

In right - triangles \(\triangle GBC\) and \(\triangle HBC\), we have \(GB = HB\) (from the property of perpendicular bisector), \(BC=BC\) (common side).
For right - triangles, if the hypotenuse and a leg are equal, the triangles are congruent. So \(\triangle GBC\cong\triangle HBC\) by HL. Then \(GC = HC\). Given \(GC = 12\), so \(HC=12\). Also, since \(AC\) is the perpendicular bisector of \(GH\), \(GB = HB\). Given \(GB = 8\), so \(HB = 8\).

Answer:

a. \(GH=GB + HB=8 + 8=16\); b. \(CH = 12\)